Zero: The Biography of a Dangerous Idea

by Charles Seife

The number of rabbits goes as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .; the number of rabbits you have in any given month is the sum of the rabbits that you had in each of the two previous months. Mathematicians instantly realized the importance of this series. Take any term and divide it by its previous term. For instance, 8/5=1.6; 13/8=1.625; 21/13=1.61538 . . . . These ratios approach a particularly interesting number: the golden ratio, which is 1.61803 . . .

Pythagoras had noticed that nature seemed to be governed by the golden ratio. Fibonacci discovered the sequence that is responsible. The size of the chambers of the nautilus and the number of clockwise grooves to counterclockwise grooves in the pineapple are governed by this sequence. This is why their ratios approach the golden ratio.

Worse yet, if the universe were infinite, then there could be no center. How could Earth, then, be the center of the universe? The answer was found in zero.

Zero and infinity were at the very center of the Renaissance. As Europe slowly awakened from the Dark Ages, the void and the infinite—nothing and everything—would destroy the Aristotelian foundation of the church and open the way to the scientific revolution.

O God, I could be bounded in a nutshell and count myself a king of infinite space, were it not that I have bad dreams. —WILLIAM SHAKESPEARE, HAMLET

Now imagine that a giant hand comes down and squashes the book flat. Instead of being a three-dimensional object, the book is now a flat, floppy rectangle. It has lost a dimension; it has length and width, but no height. It is now two-dimensional. Now imagine that the book, turned sideways, is crushed once again by the giant hand. The book is no longer a rectangle. It is a line. Again, it has lost a dimension; it has neither height nor width, but it has length. It is a one-dimensional object. You can take away even this single dimension. Squashed along its length, the line becomes a point, an infinitesimal nothing with no length, no width, and no height. A point is a zero-dimensional object.

This apparently contradictory object turned Brunelleschi’s drawing, almost magically, into such a good likeness of the three-dimensional Baptistery building that it was indistinguishable from the real thing. Indeed, when Brunelleschi used a mirror to compare the painting and the building, the reflected image matched the building’s geometry exactly. The vanishing point turned a two-dimensional drawing into a perfect simulation of a three-dimensional building.

It is no coincidence that zero and infinity are linked in the vanishing point. Just as multiplying by zero causes the number line to collapse into a point, the vanishing point has caused most of the universe to sit in a tiny dot. This is a singularity, a concept that became very important later in the history of science—but at this early stage, mathematicians knew little more than the artists about the properties of zero.

Leonardo da Vinci wrote a guide to drawing in perspective. Another of his books, about painting, warns, “Let no one who is not a mathematician read my works.”

A contemporary of Brunelleschi, a German cardinal named Nicholas of Cusa, looked at infinity and promptly declared, “Terra non est centra mundi”: the earth is not the center of the universe. The church didn’t yet realize how dangerous, how revolutionary, that idea was.

The power of Copernicus’s idea was in its simplicity. Instead of placing Earth at the center of the universe filled with epicycle-filled clockworks, Copernicus imagined that the sun was at the center instead, and the planets moved in simple circles. Planets would seem to zoom backward as Earth overtook them; no epicycles were needed. Though Copernicus’s system didn’t agree with the data completely—the circular orbits were wrong, though the heliocentric idea was correct—it was much simpler than the Ptolemaic system. The earth revolved around the sun. Terra non est centra mundi.

This was the beginning of the Reformation; intellectuals everywhere began to reject the authority of the pope.

Bruno, a former Dominican cleric, published On the Infinite Universe and Worlds, where he suggested, like Nicholas of Cusa, that the earth was not the center of the universe and that there were infinite worlds like our own. In 1600 he was burned at the stake.

In 1616 the famous Galileo Galilei, another Copernican, was ordered by the church to cease his scientific investigations. The same year, Copernicus’s De Revolutionibus was placed on the Index of forbidden books. An attack on Aristotle was considered an attack upon the church.

I am in a sense something intermediate between God and nought. —RENÉ DESCARTES, DISCOURSE ON METHOD

If nature truly abhorred a vacuum so much, the mercury in the tube would have to stay put so as not to create a void. The mercury didn’t stay put. It sank downward a bit, leaving a space at the top. What was in that space? Nothing. It was the first time in history anyone had created a sustained vacuum.

Luckily, Pascal’s lust was greater than his religious fervor for a time, because he would use science to unravel the secret of the vacuum.

It was the weight of the atmosphere pressing down on the mercury exposed in the pan that makes the fluid shoot up the column.

It is a subtle point: vacuums don’t suck; the atmosphere pushes. But Pascal’s simple experiment demolished Aristotle’s assertion that nature abhors a vacuum. Pascal wrote, “But until now one could find no one who took this . . . view, that nature has no repugnance for the vacuum, that it makes no effort to avoid it, and that it admits vacuum without difficulty and without resistance.” Aristotle was defeated, and scientists stopped fearing the void and began to study it.

What is man in nature? Nothing in relation to the infinite, everything in relation to nothing, a mean between nothing and everything. —BLAISE PASCAL, PENSÉES

Pascal was a mathematician as well as a scientist. In science Pascal investigated the vacuum—the nature of the void. In mathematics Pascal helped invent a whole new branch of the field: probability theory. When Pascal combined probability theory with zero and with infinity, he found God.

Deep within the scientific world’s powerful new tool—calculus—was a paradox. The inventors of calculus, Isaac Newton and Gottfried Wilhelm Leibniz, created the most powerful mathematical method ever by dividing by zero and adding an infinite number of zeros together. Both acts were as illogical as adding 1 + 1 to get 3. Calculus, at its core, defied the logic of mathematics. Accepting it was a leap of faith. Scientists took that leap, for calculus is the language of nature. To understand that language completely, science had to conquer the infinite zeros.

An infinite sum of numbers can be infinite, even if the numbers themselves approach zero. Yet this isn’t the strangest aspect of infinite sums. Zero itself is not immune to the bizarre nature of infinity.

In one of Kepler’s lesser-known works, Volume-Measurement of Barrels, he does this in three dimensions, slicing barrels into planes and summing the planes together. Kepler, at least, wasn’t afraid of a glaring problem: as Δx goes to zero, the sum becomes equivalent to adding an infinite number of zeros together—a result that makes no sense. Kepler ignored the problem; though adding infinite zeros together was gibberish from a logical point of view, the answer it yielded was the right one.

The telescope, for instance, had given scientists the ability to find moons and stars that had never been observed before. Calculus, on the other hand, gave scientists a way to express the laws that govern the motion of the celestial bodies—and laws that would eventually tell scientists how those moons and stars had formed.

For the truth is, nature doesn’t speak in ordinary equations. It speaks in differential equations, and calculus is the tool that you need to pose and solve these differential equations.

Differential equations are not like the everyday equations that we are all familiar with. An everyday equation is like a machine; you feed numbers into the machine and out pops another number. A differential equation is also like a machine, but this time you feed equations into the machine and out pop new equations.

Plug in an equation that describes the conditions of the problem (is the ball moving at a constant rate, or is a force acting on the ball?) and out pops the equation that encodes the answer that you seek (the ball moves in a straight line or in a parabola). One differential equation governs all of the uncountable numbers of equation-laws. And unlike the little equation-laws that sometimes hold and sometimes don’t,...

Zero and infinity always looked suspiciously alike. Multiply zero by anything and you get zero. Multiply infinity by anything and you get infinity. Dividing a number by zero yields infinity; dividing a number by infinity yields zero. Adding zero to a number leaves the number unchanged. Adding a number to infinity leaves infinity unchanged.

Zero and infinity are two sides of the same coin—equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers.

Zero is not the only number that was rejected by mathematicians for centuries. Just as zero suffered from Greek prejudice, other numbers were ignored as well, numbers that made no geometric sense. One of these numbers, i, held the key to zero’s strange properties.

Riemann merged projective geometry with the complex numbers, and all of a sudden lines became circles, circles became lines, and zero and infinity became the poles on a globe full of numbers.

(You can see this on your calculator. Enter a number—any number. Square it. Square it again. Do it again and again; the number will quickly zoom toward infinity or toward zero, except if you entered 1 or –1 to begin with. There is no escape.)

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? I have studied it. . . . I have followed its roots, so to speak, to the first infallible cause of all created things. —GEORG CANTOR

God, the infinity that defies all comprehension.

Unfortunately for Cantor, not everyone had the same vision of God. Leopold Kronecker was an eminent professor at the University of Berlin, and one of Cantor’s teachers. Kronecker believed that God would never allow such ugliness as the irrationals, much less an ever-increasing set of Russian-doll infinities. The integers represented the purity of God, while the irrationals and other bizarre sets of numbers were abominations—figments of the imperfect human mind.

Disgusted with Cantor, Kronecker launched vitriolic attacks against Cantor’s work and made it extremely difficult for him to publish papers. When Cantor applied for a position at the University of Berlin in 1883, he was rejected; he had to settle for a professorship at the much less prestigious University of Halle instead. Kronecker, who was influential at Berlin, was likely to blame. The same year, he wrote...

It would be little comfort to Cantor that his work was the foundation of a whole new branch of mathematics: set theory. Using set theory, mathematicians would not only create the numbers we know out of nothing at all, they would create numbers that were previously unheard of—infinite infinities that can be added to, multipli...

The German mathematician David Hilbert would say, “No one shall expel us from the paradise which Cantor has created for us.”

In the battle between Kronecker and Cantor, Cantor would ultimately prevail. Cantor’s theory would show that Kronecker’s precious integers—and even the rational numbers—were nothing at all. They were an infinite zero.

For instance, imagine that you’ve got a stain on your wood floor. How much area does the stain take up? It’s not so obvious. If the stain were shaped like a circle, or like a square or a triangle, it would be easy to figure out; just take a ruler and measure its radius or its height and base. But there’s no formula for figuring out the area of an amoeba-shaped mess.

Now take the second number. Cover it with a carpet of size 1/2. Take the third number and cover it with a carpet of size 1/4, and so forth. Go on and on to infinity; since every rational number is on the seating chart, every rational number will eventually be covered by a carpet. What is the total size of the carpets? It’s our old friend, the Achilles sum. Adding up the size of the carpets, we see 1 + 1/2 + 1/4 + 1/8 + . . . + 1/2n goes to 2 as n goes to infinity. So we can cover the infinite cohorts of rational numbers in the number line with a set of carpets, and the total size of the carpets is 2. This means that the rational numbers take up less than two units of space.

How big are the rational numbers? They take up no space at all. It’s a tough concept to swallow, but it’s true.

Even though there are rational numbers everywhere on the number line, they take up no space at all. If we were to throw a dart at the number line, it would never hit a rational number. Never. And though the rationals are tiny, the irrationals aren’t, since we can’t make a seating chart and cover them one by one; there will always be uncovered irrationals left over. Kronecker hated the irrationals, but they take up all the space in the number line.

Sensible mathematics involves neglecting a quantity when it is small—not neglecting it because it is infinitely great and you do not want it! —P. A. M. DIRAC

It was finally unmistakable: infinity and zero are inseparable and are essential to mathematics. Mathematicians had no choice but to learn to live with them.

For physicists, however, zero and infinity seemed utterly irrelevant to the workings of the universe. Adding infinities and dividing by zeros might be a part of mathematics, but it is not the way of nature.

As mathematicians were uncovering the connection between zero and infinity, physicists began to encounter zeros in the natural world; zero cr...

In thermodynamics a zero became an uncrossable barrier: the coldest temperature possible. In Einstein’s theory of general relativity, a zero became a black hole, a monstrous star that swallows entire suns. In quantum mechanics, a zero is responsible for a bizarre source of energy—infinite and ubiquitous, ...

When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science. —WILLIAM THOMSON, LORD KELVIN